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The effect of assuming the principal directions in neutron diffraction measurement of stress tensors
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A
Materials Science and Engineering A205 (1996) 257-258
Letter
The effect of assuming the principal directions in neutron diffraction measurement of stress tensors R.A. Winholtz, A.D. Krawitz Department of Mechanical and Aerospace Engineering and Research Reactor Center, University of Missouri, Columbia, MO 65211, USA
Received 31 July 1995
Abstract
In general, the components of stress in any three perpendicular directions can be determined by measuring the strains in those three directions. There is, however, no way to tell from the data if these are the principal stresses. The magnitude of the principal stresses will depend on the magnitude of the shear stresses that were neglected. It is shown that selecting the incorrect directions for the principal stresses leads to an error not in the components of stress measured but in the assumption that they are the principal stresses. Keywords: Stress; Diffraction
Neutron diffraction has become a useful tool for measuring stresses in the interior of materials. All six independent components of the general stress tensor can be determined using diffraction [1,2]. The determination of a general stress tensor requires measurement of the strain in at least six different orientations of the sample of interest [2,3]. If the principal stress directions are known, measurement of the strain in these three orientations will suffice to determine the stress state of the material at a given location in the material. Since neutron stress measurements can be quite time consuming, there is an incentive to presume the principal stress directions and hence reduce the measurement time. Another advantage of this method is that, for an approximately cubic gage volume, the volumes over which the strain is measured nearly overlap. A number of researchers have used this method to determine stresses and strains in components where the geometry and loading suggests the principal stress directions [4-10]. In this communication the effect of selecting the wrong principal stress directions is discussed. The fundamental relation for determining a general strain state with diffraction is [1] 0921-5093/96/$15.00 © 1996 - - Elsevier Science S.A. All rights reserved SSDI 0921-5093(95)10040-7
eoe, = s , cos 2 ~b sin 2 ~, + F~22 sin 2 q~ sin 2 ~k + s33 cos 2 + e12 sin 2~b sin 2 ¢ +/~13 COS ~b sin 2~, + ~3 sin q~ sin 2¢ Here e~e, is the strain in the direction given by the angles ~b and ~ and defined in Fig. 1 and the strain
S,
Fig. 1. Coordinate systems used in measuring stress and strain with diffraction.
258
R.A. Winholtz, A.D. Krawitz / Materials Science and Engineering A205 (1996) 257-258
components on the right-hand side of Eq. (1) are referenced to the sample coordinate system, Si, shown in Fig. 1. The strain %e, is determined by using Bragg's law to determine the interplanar spacing along the diffraction vector and using the stress-free interplanar spacing, do, to determine a strain, ~¢, = ( d - d o ) / d o. By measuring the strain, %~,, in at least six independent directions, the six components of strain in the sample coordinate system can be determined, thus characterizing the strain tensor at the measurement location [2]. The corresponding components of stress can then be determined using Hooke's law 1 [ Sl ] 0"ij = S ~ ~j - ~ij $2/2 + 3S~ Ekk
(2)
Here Sl and $2/2 are the diffraction elastic constants and 6~j is the Kr6necker delta function [1]. Note that the form of Hooke's law used assumes an isotropic material. This does not mean that the individual crystals are isotropic but that the bulk material behaves isotropically. If the principal strain axes are known, the sample coordinate system can be attached to the sample so as to coincide with them. The shear strain components, el2, el3, and F_,23, will then be zero and the principal strains, el, e2, and e3, will correspond to the strains ell, e-22, and e33, respectively, and can be determined by measuring e~e, at (q~, ¢) equal to (0,90), (90,90), and (0,0). The principal stresses are then determined using Eq. (2). One may then ask what is measured if, in fact, the principal strain (stress) directions are not in the directions assumed so t h a t the shear strain components in the sample coordinate system have non-zero values. From Eq. (1) it is seen that the shear strain components make no contribution to the measured strain %¢, at the tilts (0,90), (90,90) and (0,0). Thus, in measuring the strain in these three directions, one correctly measures these strain components no matter what the values of the shear strains. However, a possible error arises in assuming these components are the principal strains, fi, rather than the components fii- If the shear stresses are not zero the principal stresses will not be along the three sample coordinate system axes but will instead be in some other direction depending on the values of the shear strain components. The magnitude of the principal strains will depend on the magnitude of the shear
strains. The possible magnitude of the unknown principal strains is only limited by the shear strain that the material can support. Since the normal stress components do not depend on the values of the shear strains, the stress components 0-11, 0"22, and 0"33 will be correct but not necessarily equal to the principal stresses. Because the sample coordinate system in Fig. 1 may be attached to the sample in any way desired, one may generalize and say that the components of normal stress in any three perpendicular directions may be correctly determined by measuring the strains in these three directions. This is true because the shear stress components do not affect the normal strains in an isotropic material (as shown in Eq. (2)). While one may measure three perpendicular stress components with three strain measurements via diffraction, one must use other arguments, such as loading or symmetry of the sample, to infer that these are the principal stresses.
Acknowledgements This work was supported in part by the Office of Naval Research under contract N00014-93-1-1049.
References [1] I.C. Noyan and J.B. Cohen, Residual Stress: Measurement by Diffraction and Interpretation, Springer, New York, 1987. [2] R.A. Winholtz and J.B. Cohen, Aust. J. Phys., 41 (1988) 189. [3] R.A. Winholtz and A.D. Krawitz, Metall. Trans. A, 26A (1995) 1287. [4] M.J. Schmank and A.D. Krawitz, Metall. Trans. A, 13A (1982) 1069. [5] L. Pintschovius, E. Macherauch and B. Scholtes, Mater. Sci. Eng., 84 (1986) 163. [6] HJ. Prask and C.S. Choi, J. Nucl. Mater., 126 (1984) 124. [7] A.J. Allen, M.T. Hutchings, C.G. Windsor and C. Andreani, Adv. Phys., 34 (1985) 445. [8] P.J. Webster, K.S. Low, G. Mills and G.A. Webster, Mat. Res. Soc. Syrup. Proc., 166 (1990) 311. [9] T.M. Holden, R.R. Hosbons, S.R. MacEwen, E.C. Flower, M.A. Bourke and J.A. Goldstone, in M.T. Hutchings and A.D. Krawitz (eds.), Measurement of Residual and Applied Stress Using Neutron Diffraction, Ktuwer, Dordrecht, 1992, p. 93. [10] S. Spooner, X.L. Wang, C.R. Hubbard and S.A. David, Proc. Fourth Int. Conf. Residual Stresses, Society for Experimental Mechanics, Bethel, CT, 1994, p. 964.
A
Materials Science and Engineering A205 (1996) 257-258
Letter
The effect of assuming the principal directions in neutron diffraction measurement of stress tensors R.A. Winholtz, A.D. Krawitz Department of Mechanical and Aerospace Engineering and Research Reactor Center, University of Missouri, Columbia, MO 65211, USA
Received 31 July 1995
Abstract
In general, the components of stress in any three perpendicular directions can be determined by measuring the strains in those three directions. There is, however, no way to tell from the data if these are the principal stresses. The magnitude of the principal stresses will depend on the magnitude of the shear stresses that were neglected. It is shown that selecting the incorrect directions for the principal stresses leads to an error not in the components of stress measured but in the assumption that they are the principal stresses. Keywords: Stress; Diffraction
Neutron diffraction has become a useful tool for measuring stresses in the interior of materials. All six independent components of the general stress tensor can be determined using diffraction [1,2]. The determination of a general stress tensor requires measurement of the strain in at least six different orientations of the sample of interest [2,3]. If the principal stress directions are known, measurement of the strain in these three orientations will suffice to determine the stress state of the material at a given location in the material. Since neutron stress measurements can be quite time consuming, there is an incentive to presume the principal stress directions and hence reduce the measurement time. Another advantage of this method is that, for an approximately cubic gage volume, the volumes over which the strain is measured nearly overlap. A number of researchers have used this method to determine stresses and strains in components where the geometry and loading suggests the principal stress directions [4-10]. In this communication the effect of selecting the wrong principal stress directions is discussed. The fundamental relation for determining a general strain state with diffraction is [1] 0921-5093/96/$15.00 © 1996 - - Elsevier Science S.A. All rights reserved SSDI 0921-5093(95)10040-7
eoe, = s , cos 2 ~b sin 2 ~, + F~22 sin 2 q~ sin 2 ~k + s33 cos 2 + e12 sin 2~b sin 2 ¢ +/~13 COS ~b sin 2~, + ~3 sin q~ sin 2¢ Here e~e, is the strain in the direction given by the angles ~b and ~ and defined in Fig. 1 and the strain
S,
Fig. 1. Coordinate systems used in measuring stress and strain with diffraction.
258
R.A. Winholtz, A.D. Krawitz / Materials Science and Engineering A205 (1996) 257-258
components on the right-hand side of Eq. (1) are referenced to the sample coordinate system, Si, shown in Fig. 1. The strain %e, is determined by using Bragg's law to determine the interplanar spacing along the diffraction vector and using the stress-free interplanar spacing, do, to determine a strain, ~¢, = ( d - d o ) / d o. By measuring the strain, %~,, in at least six independent directions, the six components of strain in the sample coordinate system can be determined, thus characterizing the strain tensor at the measurement location [2]. The corresponding components of stress can then be determined using Hooke's law 1 [ Sl ] 0"ij = S ~ ~j - ~ij $2/2 + 3S~ Ekk
(2)
Here Sl and $2/2 are the diffraction elastic constants and 6~j is the Kr6necker delta function [1]. Note that the form of Hooke's law used assumes an isotropic material. This does not mean that the individual crystals are isotropic but that the bulk material behaves isotropically. If the principal strain axes are known, the sample coordinate system can be attached to the sample so as to coincide with them. The shear strain components, el2, el3, and F_,23, will then be zero and the principal strains, el, e2, and e3, will correspond to the strains ell, e-22, and e33, respectively, and can be determined by measuring e~e, at (q~, ¢) equal to (0,90), (90,90), and (0,0). The principal stresses are then determined using Eq. (2). One may then ask what is measured if, in fact, the principal strain (stress) directions are not in the directions assumed so t h a t the shear strain components in the sample coordinate system have non-zero values. From Eq. (1) it is seen that the shear strain components make no contribution to the measured strain %¢, at the tilts (0,90), (90,90) and (0,0). Thus, in measuring the strain in these three directions, one correctly measures these strain components no matter what the values of the shear strains. However, a possible error arises in assuming these components are the principal strains, fi, rather than the components fii- If the shear stresses are not zero the principal stresses will not be along the three sample coordinate system axes but will instead be in some other direction depending on the values of the shear strain components. The magnitude of the principal strains will depend on the magnitude of the shear
strains. The possible magnitude of the unknown principal strains is only limited by the shear strain that the material can support. Since the normal stress components do not depend on the values of the shear strains, the stress components 0-11, 0"22, and 0"33 will be correct but not necessarily equal to the principal stresses. Because the sample coordinate system in Fig. 1 may be attached to the sample in any way desired, one may generalize and say that the components of normal stress in any three perpendicular directions may be correctly determined by measuring the strains in these three directions. This is true because the shear stress components do not affect the normal strains in an isotropic material (as shown in Eq. (2)). While one may measure three perpendicular stress components with three strain measurements via diffraction, one must use other arguments, such as loading or symmetry of the sample, to infer that these are the principal stresses.
Acknowledgements This work was supported in part by the Office of Naval Research under contract N00014-93-1-1049.
References [1] I.C. Noyan and J.B. Cohen, Residual Stress: Measurement by Diffraction and Interpretation, Springer, New York, 1987. [2] R.A. Winholtz and J.B. Cohen, Aust. J. Phys., 41 (1988) 189. [3] R.A. Winholtz and A.D. Krawitz, Metall. Trans. A, 26A (1995) 1287. [4] M.J. Schmank and A.D. Krawitz, Metall. Trans. A, 13A (1982) 1069. [5] L. Pintschovius, E. Macherauch and B. Scholtes, Mater. Sci. Eng., 84 (1986) 163. [6] HJ. Prask and C.S. Choi, J. Nucl. Mater., 126 (1984) 124. [7] A.J. Allen, M.T. Hutchings, C.G. Windsor and C. Andreani, Adv. Phys., 34 (1985) 445. [8] P.J. Webster, K.S. Low, G. Mills and G.A. Webster, Mat. Res. Soc. Syrup. Proc., 166 (1990) 311. [9] T.M. Holden, R.R. Hosbons, S.R. MacEwen, E.C. Flower, M.A. Bourke and J.A. Goldstone, in M.T. Hutchings and A.D. Krawitz (eds.), Measurement of Residual and Applied Stress Using Neutron Diffraction, Ktuwer, Dordrecht, 1992, p. 93. [10] S. Spooner, X.L. Wang, C.R. Hubbard and S.A. David, Proc. Fourth Int. Conf. Residual Stresses, Society for Experimental Mechanics, Bethel, CT, 1994, p. 964.